Abstract

We study theoretically magnetohydrodynamic(MHD) motion of a binary electrolyte in a concentric annulus subjected to a uniform, axial magnetic field. The annulus’ cylindrical surfaces serve as electrodes. When a potential difference is imposed across the cylindrical electrodes, radial electric current flows in the solution and interacts with the axial magnetic field to induce a Lorentz body force that drives azimuthal fluid flow. When the annulus is infinitely long, a purely azimuthal flow (analogous to the classical Dean flow) is possible. We determine the velocity profile, ion concentration fields, and current density as functions of the electrodes’ potential difference and study the linear stability of the azimuthal flow. Of particular interest is the effect of the ions’ concentration fields on the centrifugal Dean instability. When the current is directed outwardly, electrochemical effects destabilize the flow, and the MHD flow loses stability at a Dean number much lower than its analogous, pressure driven flow. The supercritical flow consists of convective cells in the transverse plane. In contrast, when the current is directed inwardly, electrochemical effects stabilize the flow and the azimuthal flow is linearly stable for all Dean numbers. When the annulus is capped, purely azimuthal flow is no longer possible, and the flow in the annulus is always three-dimensional. In this case, the secondary flow is mostly driven by pressure gradients induced by the no-slip floor and ceiling. The intensity of the transverse convection depends then only weakly on the current's direction.

The work was supported, in part, by NSF STTR Grant No. 0822723 to SFC Fluidics.

Article outline:I. INTRODUCTIONII. MATHEMATICAL MODELIII. STEADY FLOW OF BINARY ELECTROLYTE IN AN INFINITELY LONG ANNULUS (l → +∞)IV. THE STABILITY OF THE AZIMUTHAL FLOWA. The case of controlled currentB. The case of controlled potential and Butler-Volmer boundary conditionsC. The case of controlled potential with Nernst boundary conditionsV. AN ANNULUS WITH A FINITE HEIGHT (l < ∞)A. The range of validity of the small gap approximationB. The effect of current direction on secondary convectionVI. CONCLUSIONS